AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
, not necessarily positive, we wish to select a point 
in the Cartesian plane whose coordinates lie inside the unit square defined by![\[ i < x < i+1, \qquad j < y < j+1. \]](http://latex.artofproblemsolving.com/2/3/6/2363248e26b4c9826b4f3e1794c46c358bfd8f42.png)
Find all real numbers 
for which it's possible to choose these points such that for all integers 
and 
, the (possibly concave or degenerate) quadrilateral 
has perimeter strictly less than 
.
be positive real numbers satisfying![\[ xy + yz + zx = 3xyz. \]](http://latex.artofproblemsolving.com/9/2/0/9200874fd6bf9a32ff0c4e82382a450be68cc444.png)
Prove that![\[
\sqrt{\frac{x}{3y^2z^2 + xyz}} + \sqrt{\frac{y}{3x^2z^2 + xyz}} + \sqrt{\frac{z}{3x^2y^2 + xyz}} \le \frac{3}{2}.
\]](http://latex.artofproblemsolving.com/f/5/f/f5fb85d81e93935601ceb5f30c25aea7aa3f23b2.png)
and 
be real numbers. Let 
be the matrix with entries ![\[a_{ij} = \begin{cases}1,&\text{if }x_i + y_j\geq 0;\\0,&\text{if }x_i + y_j < 0.\end{cases}\]](http://latex.artofproblemsolving.com/c/0/8/c0856fe7b2b113180659952c4d216a01d8849f64.png)
Suppose that 
is an 
matrix with entries 
, 
such that the sum of the elements in each row and each column of is equal to the corresponding sum for the matrix 
. Prove that 
.
inscribed in 
. Two points 
lie on 
such that 
are isogonal in 
. 
is the midpoint of 
. 
lies on 
such that 
. 
intersects at 
. Prove that the line through parallel to 
passes through the Euler center of triangle .
are collinear, and 
are arbitrary points
be an acute triangle and 
its orthocenter. Let be a point on the parallel through to such that 
. Define 
and 
as points on the parallels through to 
and through 
to 
similarly. If 
are positioned around the sides of as in the given configuration, prove that are collinear.
be an acute triangle with circumcenter 
. Points 
and 
are chosen on segments 
and 
such that 
. If is the midpoint of the arc 
and 
is the midpoint of the arc 
, prove that 
.
of positive integers such that ![\[\sqrt[3]{7x^2-13xy+7y^2}=|x-y|+1.\]](http://latex.artofproblemsolving.com/e/c/7/ec778edf97100f3b19350d8a817317ebf1f391bd.png)
be an integer. Ivan writes the numbers 
each on different cards. He then shuffles these 
cards, and divides them into two piles. Prove that at least one of the piles contains two cards such that the sum of their numbers is a perfect square.
integers 
, each less than or equal to 
, prove that at least one of them is divisible by some other member of the set.
be the orthocenter of acute-angled triangle 
. M is the midpoint of side BC. P is on the median AM such that 
. Prove that P lies on the circumcircle 
of triangle BHC.
.
, hence BMPE and CMPD are two cyclic quadrilateral.
Prove that
Let 
Prove that![$$ (a+b)(a+1)(b+1) \leq \frac{3}{2}+\sqrt[3]{6}+\sqrt[3]{36}$$](http://latex.artofproblemsolving.com/f/5/e/f5e68204507aa9c3e71c28844bbee3a804546b94.png)
such that![\[
f(x + y^2 + z^3) = f(x) + f(y)^2 + f(z)^3
\]](http://latex.artofproblemsolving.com/0/d/7/0d7a6f8586733073f8f543810ca00416770b84cc.png)
for all 
.
Circle 
with diameter 
and circumcircle of triangle 
intersect at point 
Prove that the straight line 
pass through the midpoint of 
